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Modeling of magnetic levitation system

Peter Balko, Danica Rosinová Institute of Automotive Mechatronics

Slovak University of Technology in Bratislava Bratislava, Slovakia

[email protected], [email protected]

Abstract—The paper is devoted to modeling and parameter identification for Magnetic Levitation System (MLS) from Inteco. MLS belongs to challenging modelling and control problems due to its unstability and nonlinearity. We concern several modeling details not sufficiently described in user manual, a correction of nonlinear model, and present the corresponding measurement results. The obtained nonlinear model parameters and corresponding linearized model fits the real data much better than parameters provided in the reference.

Keywords—nonlinear system, modeling, unstable system,

magnetic levitation (maglev)

I. INTRODUCTION Magnetic levitation belongs to well-known type of systems

with variety of application field, [1]-[4]. Modeling the magnetic levitation system belongs to challenging tasks, since besides nonlinearities we have to cope with open-loop unstable system with fast dynamics and very small degree of natural damping. The basic control aim is to precisely position the levitating object, which requires adequately precise model. There exist many references devoted to modelling and control of various types of magnetic levitation laboratory plants, [5]-[8], to list a few. The laboratory MLS provided by Inteco is analyzed e.g. in [9], [11], [12], however, to the authors best knowledge, several important modeling details have not been sufficiently reported.

Basically, a mathematical model can be obtained from basic physical laws (first principles) or by data driven identification methods based on measurement of input and output data for adequately excited system, [10]. Frequently, both these approaches are combined, the structure is determined according to a theoretical (first principle based) model and parameter values are estimated from measured data using identification methods.

In this paper, we present our results on modeling the laboratory magnetic levitation system [7] with 2 electromagnets. The controlled output is a ball position between the electromagnets, where we consider only the upper one.

We present a model based on first principles, with simplified nonlinearities, where the respective parameters are estimated from measured data and verified by simulations and the resulting model is compared with experimental results on real laboratory plant.

Fig. 1. Magnetic Levitation System with 2 Electromagnets EM1 and EM2 and a levitating sphere ball between them.

II. MAGNETIC LEVITATION SYSTEMS: PHYSICAL MODEL Magnetic levitation can be briefly characterized as

achieving the equilibrium of an object in the air-space without contact with solid substance, by using electromagnetic forces. In MLS, Fig. 1, a ferromagnetic object (a ball) is kept in the air-space between two electromagnets, where the upper one provides a vertical force overcoming the ball gravity and the lower one is used as additional, mainly for stabilization of a ball horizontal position.

A. Development of a Magnetic levitation model

The nonlinear physical model of the MLS can be described by state equations obtained from basic physical laws for a ball (sphere) motion in electromagnetic field, [9], [11].

Description of a ball dynamics and of electromagnetic forces is based on Lagrange function - the difference between kinetic and potential energy, which can be written as

t

qumgxdtqRqxLxmT0

222

21)(

21

21

where x is a distance of the sphere from electromagnet, q is electric charge, m is a mass of the sphere, R is a resistance of the electromagnet coil, L(x) is a function describing the

2017 21st International Conference on Process Control (PC)June 6–9, 2017, Štrbské Pleso, Slovakia

978-1-5386-4011-1/17/$31.00 c©2017 IEEE 252

dependence of inductance of the coil on distance x, qI is a current in the coil, g is gravity constant, u is voltage.

Variables )(),( tqtx must meet the Lagrange equations

0

0

q

T

qd

dT

dt

d

x

T

xd

dT

dt

d

which yield

uRIIdt

dx

dx

dL

Ldt

dI

gIdx

dL

mdt

xd

1

,21 2

2

2

The first equation from (3) corresponds to Newton’s second law, where an electromagnetic force is given by

221),( I

dx

dLIxF

Dependence of coil inductance on distance can be approximated by polynomial or exponential functions

0,)( 10 aeLLxL ax a

Raan

axaxaxaLxL

n

n

n

n

n

,...,,1

,...

1)(

0

0111

10 b

Below, we will use the exponential alternative (5a), having

axeaLdx

dL 1

The second equation from (3) is simplified by the experimentally received approximation, [9]

)()(

1Icku

xfdt

dI

where function )(xf has a structure similar to (6). Introducing state variables

,sphere theofposition )()(1 txtx,sphere theof velocity )()( 12 txtx

coil,upper in thecurrent )()(3 tItx

and combining (3) and (7), the resulting nonlinear mathematical model is obtained, see [5,9]

)()(

1

)(21

,

31

3

23

1

12

21

xckuxfdt

dx

gxdx

xdL

mdt

dx

xdt

dx

Applying the approximation (6) with slightly different denotation, finally we obtain

)()(

12

,

31

3

12

21

xcukxfdt

dx

gm

F

dt

dx

xdt

dx

ii

i

em

where

.exp

,exp

2

1

2

11

2

1

2

1231

iPiP

iPi

emPemP

emPem

f

x

f

fxf

F

x

F

FxF

Correspondence of the second equalities from (8) and (9) is through approximation (6) with

2

1

2

111

2

exp,,1

emPemP

emPemP

emP F

x

F

F

dx

dLFL

Fa

The real laboratory model uses variables in the intervals

1,,38.2,,,016.0,0 minmin321 uuixRxx .

Remark 1: It should be noted that the second equation in nonlinear model (9) differs from the one presented in MLS documentation, [5]: there is coefficient 2 in the denominator corresponding to the first equation from (3). This coefficient appears then also in the linearized model below (in elements

1,2a and 3,2a in (13)).

B. Validation of model coefficients

This section presents validation of the key coefficients from nonlinear model (9) by measurements on laboratory plant. Since the plant itself is unstable, measurements are realized in closed-loop with stabilizing PD controller delivered by producer, [5].

253

position [mm]8 9 10 11 12 13 14

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

measured dataour approximationapproximation from literature

Fig. 2. Validation of approximation coefficients, obtained from measured dependence of coil current I on ball position x.

Coefficients 1emPF and 2emPF for approximation of dxdL / corresponding to (6), can be obtained by measuring the

dependence of coil current 3x on ball position 1x in steady state. dxdL / is then computed from the second equality from (8)

231

1

23

1

1

2)(

)(210

x

mg

dx

xdL

gxdx

xdL

m

Measured data 1x , 3x and u are given in Tab.I. The corresponding Fig.2 shows comparison of approximation for coefficients 1emPF and 2emPF obtained from measured data, and the coefficients reported in Inteco manual. We can confirm near accordance of measured and reported approximation coefficients.

TABLE I. MEASURED x1, x3 AND u FOR dL/dx1 APPROXIMATION

Position (x1) [mm]

Current (x3) [A]

Control output (u) [MU]

-dL/dx1 [H/m]

8.5 1.025 0.315 0.7096 9 1.078 0.328 0.6416 9.5 1.126 0.344 0.588 10 1.189 0.352 0.5274 10.5 1.237 0.362 0.4872 11 1.285 0.378 0.4515 11.5 1.35 0.392 0.4091 12 1.408 0.413 0.3761 12.5 1.475 0.423 0.3427 13 1.543 0.432 0.3131 13.5 1.617 0.446 0.2851 14 1.679 0.462 0.2645

All parameters of the above equations are given in Table II.

TABLE II. PARAMETERS OF THE EQUATIONS (1)

Parameters Values Units m 0.016, 0.023, 0.039 kg g 9.81 m/s2 Fem1,Fem2 function of x1 and x3 N FemP1 0.017521 H FemP2 0.0058231 m fi(x1) function of x1 s-1 fiP1 1.4142 10-4 ms fiP2 4.5626 10-3 m ci -0.4 A ki 4.4 A xd distance between electromagnets minus ball

diameter (0.1- 0.033 / 0.04 / 0.054) m

iMIN 0.03884 A uMIN 0.00498 MU

control output [MU]0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

measured dataaproximation data

Fig. 3. Dependence of current x3 on control output (provided in machine units) for all three balls

Parameters ki and ci are given by approximation of real measured data, see Fig.3. We made more experiments for all three balls, the average parameter values are

4.04.4 ucukI ii

The difference between real data and approximation is very small, thus we can consider the corresponding nonlinear model as appropriate for linearization around equilibrium points. In this case, the obtained parameters ki and ci are significantly different from those reported in documentation, [5].

III. LINEARISATION OF MAGNETIC LEVITATION SYSTEM The linearized state space model for (1) can be obtained

using Jacobian linearization around the determined working point as

txtCxty

tu

b

tx

a

aatButAxtx

001

00

000

010

33,3

3,21,2

254

where the elements of the A and B matrices are for the

working point defined by [ ],, 302010 xxx , given as

103

103,3

2

10

2

1303,2

2

102

2

1230

1,2

1

1

exp

exp2

xfkb

xfa

F

x

F

F

m

xa

F

x

F

F

m

xa

i

i

i

emPemP

emP

emPemP

emP

Note that 01,3 a since 03 xcuk ii in any working point.

To convert state space model to transfer function, we use 0)det( AsI

3,3

3,21,2

00

01)det(

as

asa

s

AsI

3,31,21,2

3,31,22det

asasas

asasAsI

The resulting transfer function is

3,31,223,231

)()()(

asas

abBAsIC

sU

sYsG

T

The input current respective to the position of the ball is depicted in Fig. 4 for all three balls. This dependence is used for determining the working point corresponding to the required ball position.

In the following we present three linearized models, where parameters 030 ,, uxm vary for big, medium and small balls.

Determination of the working point

The working point is determined by a required ball position 10x . Thus the working point is set by the next steps.

i) choose the ball size m and its position 10x ; ii) set the value of 30x corresponding to the determined 10x

(see Fig.4.), alternatively – calculate 30x from the second equation from (9) in steady state for 101 xx ;

iii) calculate u0 for the determined 10x and 30x from (11): 4.04.4 0030 ucukx ii .

Fig. 4. Dependence of coil current on ball position

Big ball linearized model

We consider the working point ][01.010 mx , ][19.130 Ax and ][363.010 MUu , the state space model is

txtCxty

tutxtButAxtx

0011271

00

288.8-0016.5-01685010

and the respective transfer function is

04.4104.418.28820947

1

sss

sG

Medium ball linearized model

We consider the working point ][01.010 mx , ][914.030 Ax and ][300.010 MUu , the state space model is

txtCxty

tutxtButAxtx

0011271

00

288.8-0021.5-01685010

and the respective transfer function is

04.4104.418.28827277

2

sss

sG

Small ball linearized model

We consider the working point ][01.010 mx , ][762.030 Ax and ][265.010 MUu

255

txtCxty

tutxtButAxtx

0011271

00

288.8-0025.7-01685010

and the respective transfer function is

04.4104.418.28832704

3

sss

sG

Note that only gain varies with the ball change.

Comparison between real and linearised system

In this section, step responses around working points for all three balls are shown for:

1) real laboratory MLS;

2) nonlinear model (9) with parameter values given in Tab.II, denoted as nonlin. model 1;

3) nonlinear model with parameters from [5], denoted as nonlin. model 2;

4) linearized models (16), (17) and (18) for big, medium and small ball respectively.

The MLS system is unstable. Therefore the comparison is realized in closed loop with stabilizing controller. In our case, we use a PID controller corresponding to (20) with parameters

P=125; I=377 830; D=5.65.

(Note that sampling time for MLS is 1ms.)

The results are illustrated in the next pictures. Comparison shows that the developed nonlin. model 1 as well as the corresponding linearized model better fits the real MLS response than nonlin. model 2 reported in [5]. (Reason can be in slightly different parameters of the individual MLSs.)

18 20 22 24 26 28 307.5

8

8.5

9

9.5

10

10.5

11

11.5

12small ball

dist

ance

[mm

]

time [s]

real datanonlin. model 1nonlin. model 2linearized model

Fig. 5. Comparison of the output responses of the developed model, linearized one, nonlinear model from [5] and real measured data for small ball

18 20 22 24 26 28 307.5

8

8.5

9

9.5

10

10.5

11

11.5

12medium ball

dist

ance

[mm

]

time [s]

real datanonlin. model 1nonlin. model 2linearized model

Fig. 6. Comparison of the output responses of the developed model, linearized one, nonlinear model from [5] and real measured data for medium ball

18 20 22 24 26 28 307.5

8

8.5

9

9.5

10

10.5

11

11.5

12big ball

dist

ance

[mm

]

time [s]

real datanonlin. model 1nonlin. model 2linearized model

Fig. 7. Comparison of the output responses of the developed model, linearized one, nonlinear model from [5] and real measured data for big ball

It is important to note that though thesystem is highly nonlinear, the linearized model works well around the working point.

IV. STABILITY CONDITIONS FOR A CLOSED LOOP WITH PID CONTROLLER

In this section, necessary stability conditions are developed for the closed loop comprising linearized MLS model with PID controller, which indicates the required controller structure and parameters.

Due to negative gain of the MLS transfer function (16), (17), (18), positive feedback is considered to obtain control error. The PID controller design is then based on closed loop

256

sGsGsGsG

sGPID

PIDCL

1

where sG correspond to MLS transfer function (15), its parameters are determined by working point and respective ball (see (16), (17), (18)); sGPID is transfer function of PID controller in the form

Dss

IPsGPID

Closed-loop characteristic polynomial for (19) with sG and sGPID given by (15) and (20) is

IabPabaasabasasCHP 3,233,233,31,22

3,231,23

3,34 )()(

Recall the parameter signs (compare (15) with e.g. (16))

0;0;0;0 33,33,21,2 baaa , therefore

0;0 3,233,31,2 abaa

Applying Routh stability criterion on (21) we receive closed loop stability conditions

3,23

3,31,2

3,23

1,2 ;ab

aaP

ab

aD

Therefore

3,3a

PD

Stability analysis shows that minimal stabilizing structure comprises proportional P and derivative D term. To avoid steady-state error, integral term is required as well.

Controller design procedure is not included in this paper, one possible control design procedure is described in [14].

V. CONCLUSION We have presented several details on modelling Magnetic

levitation system delivered by Inteco, not reported in the system documentation, and corrected a mistake in theoretical model. The experiments to verify values of model parameters

are described in details; the respective results show the major difference in linear approximation of dependence of coil current on input voltage in comparison with [5]. Modifying the model parameter values according to measured data provides nonlinear and respective linearized model with step response very close to the real experiments.

ACKNOWLEDGMENT

The work has been supported by the SRDA grant No APVV 0772-12 and Slovak Scientific Grant Agency, grant No 1/0733/16.

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